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MotivationBasic Modal LogicLogic Engineering
08—Modal Logic
CS 3234: Logic and Formal Systems
Martin Henz and Aquinas Hobor
October 7, 2010
Generated on Thursday 7th October, 2010, 11:34
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
1 Motivation
2 Basic Modal Logic
3 Logic Engineering
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
1 Motivation
2 Basic Modal Logic
3 Logic Engineering
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Necessity
You are crime investigator and consider different suspects.You know that the victim Ms Smith had called the police.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Necessity
You are crime investigator and consider different suspects.You know that the victim Ms Smith had called the police.
Maybe the cook did it before dinner?
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Necessity
You are crime investigator and consider different suspects.You know that the victim Ms Smith had called the police.
Maybe the cook did it before dinner?Maybe the maid did it after dinner?
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Necessity
You are crime investigator and consider different suspects.You know that the victim Ms Smith had called the police.
Maybe the cook did it before dinner?Maybe the maid did it after dinner?
But: “The victim Ms Smith made a phone call before shewas killed.” is necessarily true.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Necessity
You are crime investigator and consider different suspects.You know that the victim Ms Smith had called the police.
Maybe the cook did it before dinner?Maybe the maid did it after dinner?
But: “The victim Ms Smith made a phone call before shewas killed.” is necessarily true.
“Necessarily” means in all possible scenarios (worlds)under consideration.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Notions of Truth
Often, it is not enough to distinguish between “true” and“false”.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Notions of Truth
Often, it is not enough to distinguish between “true” and“false”.We need to consider modalities if truth, such as:
necessity (“in all possible scenarios”)morality/law (“in acceptable/legal scenarios”)time (“forever in the future”)
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Notions of Truth
Often, it is not enough to distinguish between “true” and“false”.We need to consider modalities if truth, such as:
necessity (“in all possible scenarios”)morality/law (“in acceptable/legal scenarios”)time (“forever in the future”)
Modal logic constructs a framework using which modalitiescan be formalized and reasoning methods can beestablished.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
1 Motivation
2 Basic Modal LogicSyntaxSemanticsEquivalences
3 Logic Engineering
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Syntax of Basic Modal Logic
φ ::= ⊤ | ⊥ | p | (¬φ) | (φ ∧ φ)| (φ ∨ φ) | (φ→ φ)
| (�φ) | (♦φ)
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Pronunciation and Examples
Pronunciation
If we want to keep the meaning open, we simply say “box” and“diamond”.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Pronunciation and Examples
Pronunciation
If we want to keep the meaning open, we simply say “box” and“diamond”.If we want to appeal to our intuition, we may say “necessarily”and “possibly” (or “forever in the future” and “sometime in thefuture”)
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Pronunciation and Examples
Pronunciation
If we want to keep the meaning open, we simply say “box” and“diamond”.If we want to appeal to our intuition, we may say “necessarily”and “possibly” (or “forever in the future” and “sometime in thefuture”)
Examples
(p ∧ ♦(p → �¬r))
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Pronunciation and Examples
Pronunciation
If we want to keep the meaning open, we simply say “box” and“diamond”.If we want to appeal to our intuition, we may say “necessarily”and “possibly” (or “forever in the future” and “sometime in thefuture”)
Examples
(p ∧ ♦(p → �¬r))
�((♦q ∧ ¬r) → �p)
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Kripke Models
Definition
A model M of propositional modal logic over a set ofpropositional atoms A is specified by three things:
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Kripke Models
Definition
A model M of propositional modal logic over a set ofpropositional atoms A is specified by three things:
1 A W of worlds;
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Kripke Models
Definition
A model M of propositional modal logic over a set ofpropositional atoms A is specified by three things:
1 A W of worlds;2 a relation R on W , meaning R ⊆ W × W , called the
accessibility relation;
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Kripke Models
Definition
A model M of propositional modal logic over a set ofpropositional atoms A is specified by three things:
1 A W of worlds;2 a relation R on W , meaning R ⊆ W × W , called the
accessibility relation;3 a function L : W → A → {T ,F}, called labeling function.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Who is Kripke?
How do I know I am not dreaming? Saul Kripke asked himselfthis question in 1952, at the age of 12.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Who is Kripke?
How do I know I am not dreaming? Saul Kripke asked himselfthis question in 1952, at the age of 12. His fathertold him about the philosopher Descartes.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Who is Kripke?
How do I know I am not dreaming? Saul Kripke asked himselfthis question in 1952, at the age of 12. His fathertold him about the philosopher Descartes.
Modal logic at 17 Kripke’s self-studies in philosophy and logicled him to prove a fundamental completenesstheorem on modal logic at the age of 17.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Who is Kripke?
How do I know I am not dreaming? Saul Kripke asked himselfthis question in 1952, at the age of 12. His fathertold him about the philosopher Descartes.
Modal logic at 17 Kripke’s self-studies in philosophy and logicled him to prove a fundamental completenesstheorem on modal logic at the age of 17.
Bachelor in Mathematics from Harvard is his onlynon-honorary degree
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Who is Kripke?
How do I know I am not dreaming? Saul Kripke asked himselfthis question in 1952, at the age of 12. His fathertold him about the philosopher Descartes.
Modal logic at 17 Kripke’s self-studies in philosophy and logicled him to prove a fundamental completenesstheorem on modal logic at the age of 17.
Bachelor in Mathematics from Harvard is his onlynon-honorary degree
At Princeton Kripke taught philosophy from 1977 onwards.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Who is Kripke?
How do I know I am not dreaming? Saul Kripke asked himselfthis question in 1952, at the age of 12. His fathertold him about the philosopher Descartes.
Modal logic at 17 Kripke’s self-studies in philosophy and logicled him to prove a fundamental completenesstheorem on modal logic at the age of 17.
Bachelor in Mathematics from Harvard is his onlynon-honorary degree
At Princeton Kripke taught philosophy from 1977 onwards.
Contributions include modal logic, naming, belief, truth, themeaning of “I”
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Example
W = {x1, x2, x3, x4, x5, x6}R = {(x1, x2), (x1, x3), (x2, x2), (x2, x3), (x3, x2), (x4, x5), (x5, x4), (x5, x6)}L = {(x1, {q}), (x2, {p, q}), (x3, {p}), (x4, {q}), (x5, {}), (x6, {p})}
pq
p
q
p, q
x1
x2
x3
x4
x5
x6
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R, L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R, L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
x ⊤
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R, L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
x ⊤x 6 ⊥
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R, L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
x ⊤x 6 ⊥x p iff L(x)(p) = T
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R, L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
x ⊤x 6 ⊥x p iff L(x)(p) = T
x ¬φ iff x 6 φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R, L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
x ⊤x 6 ⊥x p iff L(x)(p) = T
x ¬φ iff x 6 φ
x φ ∧ ψ iff x φ and x ψ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R, L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
x ⊤x 6 ⊥x p iff L(x)(p) = T
x ¬φ iff x 6 φ
x φ ∧ ψ iff x φ and x ψ
x φ ∨ ψ iff x φ or x ψ
...
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition (continued)
Let M = (W ,R, L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
...
x φ→ ψ iff x ψ, whenever x φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition (continued)
Let M = (W ,R, L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
...
x φ→ ψ iff x ψ, whenever x φ
x �φ iff for each y ∈ W with R(x , y), we have y φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition (continued)
Let M = (W ,R, L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
...
x φ→ ψ iff x ψ, whenever x φ
x �φ iff for each y ∈ W with R(x , y), we have y φ
x ♦φ iff there is a y ∈ W such that R(x , y) and y φ.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Example
pq
p
q
p, q
x1
x2
x3
x4
x5
x6
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Example
pq
p
q
p, q
x1
x2
x3
x4
x5
x6
x1 q
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Example
pq
p
q
p, q
x1
x2
x3
x4
x5
x6
x1 q
x1 ♦q, x1 6 �q
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Example
pq
p
q
p, q
x1
x2
x3
x4
x5
x6
x1 q
x1 ♦q, x1 6 �q
x5 6 �p, x5 6 �q, x5 6 �p ∨�q, x5 �(p ∨ q)
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Example
pq
p
q
p, q
x1
x2
x3
x4
x5
x6
x1 q
x1 ♦q, x1 6 �q
x5 6 �p, x5 6 �q, x5 6 �p ∨�q, x5 �(p ∨ q)
x6 �φ holds for all φ, but x6 6 ♦φ regardless of φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Formula Schemes
Example
We said x6 �φ holds for all φ, but x6 6 ♦φ regardless of φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Formula Schemes
Example
We said x6 �φ holds for all φ, but x6 6 ♦φ regardless of φ
Notation
Greek letters denote formulas, and are not propositional atoms.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Formula Schemes
Example
We said x6 �φ holds for all φ, but x6 6 ♦φ regardless of φ
Notation
Greek letters denote formulas, and are not propositional atoms.
Formula schemes
Terms where Greek letters appear instead of propositionalatoms are called formula schemes.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Entailment and Equivalence
Definition
A set of formulas Γ entails a formula ψ of basic modal logic if, inany world x of any model M = (W ,R, L), whe have x ψ
whenever x φ for all φ ∈ Γ. We say Γ entails ψ and writeΓ |= ψ.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Entailment and Equivalence
Definition
A set of formulas Γ entails a formula ψ of basic modal logic if, inany world x of any model M = (W ,R, L), whe have x ψ
whenever x φ for all φ ∈ Γ. We say Γ entails ψ and writeΓ |= ψ.
Equivalence
We write φ ≡ ψ if φ |= ψ and ψ |= φ.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Some Equivalences
De Morgan rules: ¬�φ ≡ ♦¬φ, ¬♦φ ≡ �¬φ.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Some Equivalences
De Morgan rules: ¬�φ ≡ ♦¬φ, ¬♦φ ≡ �¬φ.
Distributivity of � over ∧:
�(φ ∧ ψ) ≡ �φ ∧�ψ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Some Equivalences
De Morgan rules: ¬�φ ≡ ♦¬φ, ¬♦φ ≡ �¬φ.
Distributivity of � over ∧:
�(φ ∧ ψ) ≡ �φ ∧�ψ
Distributivity of ♦ over ∨:
♦(φ ∨ ψ) ≡ ♦φ ∨ ♦ψ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Some Equivalences
De Morgan rules: ¬�φ ≡ ♦¬φ, ¬♦φ ≡ �¬φ.
Distributivity of � over ∧:
�(φ ∧ ψ) ≡ �φ ∧�ψ
Distributivity of ♦ over ∨:
♦(φ ∨ ψ) ≡ ♦φ ∨ ♦ψ
�⊤ ≡ ⊤, ♦⊥ ≡ ⊥
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Validity
Definition
A formula φ is valid if it is true in every world of every model, i.e.iff |= φ holds.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Examples of Valid Formulas
All valid formulas of propositional logic
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Examples of Valid Formulas
All valid formulas of propositional logic
¬�φ→ ♦¬φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Examples of Valid Formulas
All valid formulas of propositional logic
¬�φ→ ♦¬φ�(φ ∧ ψ) → �φ ∧�ψ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Examples of Valid Formulas
All valid formulas of propositional logic
¬�φ→ ♦¬φ�(φ ∧ ψ) → �φ ∧�ψ
♦(φ ∨ ψ) → ♦φ ∨ ♦ψ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Examples of Valid Formulas
All valid formulas of propositional logic
¬�φ→ ♦¬φ�(φ ∧ ψ) → �φ ∧�ψ
♦(φ ∨ ψ) → ♦φ ∨ ♦ψ
Formula K : �(φ→ ψ) → �φ→ �ψ.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
1 Motivation
2 Basic Modal Logic
3 Logic EngineeringValid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
It will always be true that φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
It will always be true that φ
It ought to be that φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
It will always be true that φ
It ought to be that φ
Agent Q believes that φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
It will always be true that φ
It ought to be that φ
Agent Q believes that φ
Agent Q knows that φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
It will always be true that φ
It ought to be that φ
Agent Q believes that φ
Agent Q knows that φ
After any execution of program P, φ holds.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
It will always be true that φ
It ought to be that φ
Agent Q believes that φ
Agent Q knows that φ
After any execution of program P, φ holds.
Since ♦φ ≡ ¬�¬φ, we can infer the meaning of ♦ in eachcontext.
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φAgent Q believes that φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φAgent Q believes that φ φ is consistent with Q’s beliefs
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φAgent Q believes that φ φ is consistent with Q’s beliefsAgent Q knows that φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φAgent Q believes that φ φ is consistent with Q’s beliefsAgent Q knows that φ For all Q knows, φ
CS 3234: Logic and Formal Systems 08—Modal Logic
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φAgent Q believes that φ φ is consistent with Q’s beliefsAgent Q knows that φ For all Q knows, φAfter any run of P, φ holds.
CS 3234: Logic and Formal Systems 08—Modal Logic
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A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φAgent Q believes that φ φ is consistent with Q’s beliefsAgent Q knows that φ For all Q knows, φAfter any run of P, φ holds. After some run of P, φ holds
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Formula Schemes that hold wrt some Modalities
�φ �φ→φ
�φ→
��φ
♦φ→
�♦φ
♦⊤ �φ→
♦φ
�φ∨�
¬φ
�(φ→ψ)
∧�φ→
�ψ
♦φ∧ ♦ψ→
♦(φ∧ ψ
)
It is necessary that φ√ √ √ √ √ × √ ×
It will always be that φ × √ × × × × √ ×It ought to be that φ × × × √ √ × √ ×Agent Q believes that φ × √ √ √ √ × √ ×Agent Q knows that φ
√ √ √ √ √ × √ ×After running P, φ × × × × × × √ ×
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Modalities lead to Interpretations of R�φ R(x , y)
It is necessarily true that φ y is possible world according to info at x
It will always be true that φ y is a future world of x
It ought to be that φ y is an acceptable world according to theinformation at x
Agent Q believes that φ y could be the actual world according toQ’s beliefs at x
Agent Q knows that φ y could be the actual world according toQ’s knowledge at x
After any execution of P, φholds
y is a possible resulting state after execu-tion of P at x
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).transitive: for every x , y , z ∈ W , we have R(x , y) andR(y , z) imply R(x , z).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).transitive: for every x , y , z ∈ W , we have R(x , y) andR(y , z) imply R(x , z).Euclidean: for every x , y , z ∈ W with R(x , y) and R(x , z),we have R(y , z).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).transitive: for every x , y , z ∈ W , we have R(x , y) andR(y , z) imply R(x , z).Euclidean: for every x , y , z ∈ W with R(x , y) and R(x , z),we have R(y , z).functional: for each x there is a unique y such that R(x , y).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).transitive: for every x , y , z ∈ W , we have R(x , y) andR(y , z) imply R(x , z).Euclidean: for every x , y , z ∈ W with R(x , y) and R(x , z),we have R(y , z).functional: for each x there is a unique y such that R(x , y).linear: for every x , y , z ∈ W with R(x , y) and R(x , z), wehave R(y , z) or y = z or R(z, y).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).transitive: for every x , y , z ∈ W , we have R(x , y) andR(y , z) imply R(x , z).Euclidean: for every x , y , z ∈ W with R(x , y) and R(x , z),we have R(y , z).functional: for each x there is a unique y such that R(x , y).linear: for every x , y , z ∈ W with R(x , y) and R(x , z), wehave R(y , z) or y = z or R(z, y).total: for every x , y ∈ W , we have R(x , y) and R(y , x).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).transitive: for every x , y , z ∈ W , we have R(x , y) andR(y , z) imply R(x , z).Euclidean: for every x , y , z ∈ W with R(x , y) and R(x , z),we have R(y , z).functional: for each x there is a unique y such that R(x , y).linear: for every x , y , z ∈ W with R(x , y) and R(x , z), wehave R(y , z) or y = z or R(z, y).total: for every x , y ∈ W , we have R(x , y) and R(y , x).equivalence: reflexive, symmetric and transitive.
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Example
Consider the modality in which �φ means“it ought to be that φ”.
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Example
Consider the modality in which �φ means“it ought to be that φ”.
Should R be reflexive?
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Example
Consider the modality in which �φ means“it ought to be that φ”.
Should R be reflexive?
Should R be serial?
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Necessarily true and Reflexivity
Guess
R is reflexive if and only if �φ→ φ is valid.
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Motivation
We would like to establish that some formulas holdwhenever R has a particular property.
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Motivation
We would like to establish that some formulas holdwhenever R has a particular property.
Ignore L, and only consider the (W ,R) part of a model,called frame.
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Motivation
We would like to establish that some formulas holdwhenever R has a particular property.
Ignore L, and only consider the (W ,R) part of a model,called frame.
Establish formula schemes based on properties of frames.
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Reflexivity and Transitivity
Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
R is reflexive;
F satisfies �φ→ φ;
F satisfies �p → p for any atom p
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Reflexivity and Transitivity
Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
R is reflexive;
F satisfies �φ→ φ;
F satisfies �p → p for any atom p
Theorem 2
The following statements are equivalent:
R is transitive;
F satisfies �φ→ ��φ;
F satisfies �p → ��p for any atom p
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
1 ⇒ 2: Let R be reflexive.
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
1 ⇒ 2: Let R be reflexive. Let L be any labeling function;M = (W ,R, L).
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
1 ⇒ 2: Let R be reflexive. Let L be any labeling function;M = (W ,R, L). Need to show for any x :x �φ→ φ
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
1 ⇒ 2: Let R be reflexive. Let L be any labeling function;M = (W ,R, L). Need to show for any x :x �φ→ φ Suppose x �φ.
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
1 ⇒ 2: Let R be reflexive. Let L be any labeling function;M = (W ,R, L). Need to show for any x :x �φ→ φ Suppose x �φ.Since R is reflexive, we have x φ.
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
1 ⇒ 2: Let R be reflexive. Let L be any labeling function;M = (W ,R, L). Need to show for any x :x �φ→ φ Suppose x �φ.Since R is reflexive, we have x φ.Using the semantics of →: x �φ→ φ
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
2 ⇒ 3: Just set φ to be p
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
3 ⇒ 1: Suppose the frame satisfies �p → p.
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
3 ⇒ 1: Suppose the frame satisfies �p → p.Take any world x from W .
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
3 ⇒ 1: Suppose the frame satisfies �p → p.Take any world x from W .Choose a labeling function L such thatL(x)(p) = F , but L(y)(p) = T for all y with y 6= x
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
3 ⇒ 1: Suppose the frame satisfies �p → p.Take any world x from W .Choose a labeling function L such thatL(x)(p) = F , but L(y)(p) = T for all y with y 6= xProof by contradiction: Assume (x , x) 6∈ R.
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
3 ⇒ 1: Suppose the frame satisfies �p → p.Take any world x from W .Choose a labeling function L such thatL(x)(p) = F , but L(y)(p) = T for all y with y 6= xProof by contradiction: Assume (x , x) 6∈ R. Thenwe would have x �p, but not x p.Contradiction!
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Formula Schemes and Properties of R
name formula scheme property of RT �φ→ φ reflexiveB φ→ �♦φ symmetricD �φ→ ♦φ serial4 �φ→ ��φ transitive5 ♦φ→ �♦φ Euclidean
�φ→ ♦φ ∧ ♦φ→ �φ functional�(φ ∧�φ→ ψ) ∨�(ψ ∧�ψ → φ) linear
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Which Formula Schemes to Choose?
Definition
Let L be a set of formula schemes and Γ ∪ {ψ} a set offormulas of basic modal logic.
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Which Formula Schemes to Choose?
Definition
Let L be a set of formula schemes and Γ ∪ {ψ} a set offormulas of basic modal logic.
A set of formula schemes is said to be closed iff it containsall substitution instances of its elements.
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Which Formula Schemes to Choose?
Definition
Let L be a set of formula schemes and Γ ∪ {ψ} a set offormulas of basic modal logic.
A set of formula schemes is said to be closed iff it containsall substitution instances of its elements.
Let Lc be the smallest closed superset of L.
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Which Formula Schemes to Choose?
Definition
Let L be a set of formula schemes and Γ ∪ {ψ} a set offormulas of basic modal logic.
A set of formula schemes is said to be closed iff it containsall substitution instances of its elements.
Let Lc be the smallest closed superset of L.
Γ entails ψ in L iff Γ ∪ Lc semantically entails ψ. We sayΓ |=L ψ.
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Examples of Modal Logics: K
K is the weakest modal logic, L = ∅.
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Examples of Modal Logics: KT45
L = {T , 4, 5}
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Examples of Modal Logics: KT45
L = {T , 4, 5}
Used for reasoning about knowledge.
T: Truth: agent Q only knows true things.
4: Positive introspection: If Q knows something, he knowsthat he knows it.
5: Negative introspection: If Q doesn’t know something, heknows that he doesn’t know it.
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Next Week
Examples of Modal Logic
Natural deduction in modal logic
Modal logic in Coq
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